ALLIGATION ALTERNATE teaches the method of finding what proportion must be taken of several simples, whose prices are known, to form a compound of a given price.

Alligation Alternate is the reverse of Alligation Medial, and may be proved by it.

For a first example, let us take the one above stated. If oats worth 3s. per bushel be mixed with wheat worth 9s., how much must be taken of each sort that the compound may be worth 5s. per bushel?

3

4 Oats.

5

9

2 Wheat.

If the price of the mixture were 6s., half the sum of the prices of the simples, it is plain that it would be necessary to take just as much oats as wheat.

But since the price of the mixture is nearer to the price of the oats than to that of the wheat, less wheat will be required in the mixture than oats.

Having set down the prices of the simples under each other, and linked them together, we next set 5s., the price of the mixture, on the left. We then take the difference between 9 and 5 and place it opposite 3, the price of the oats, and also the difference between 5 and 3, and place it opposite 9, the price of the wheat. The difference standing opposite each kind shows how much of that kind is to be taken. In the present example, the mixture will consist of 4 bushels of oats and 2 of wheat; and any other quantities, bearing the same proportion to each other, such as 8 and 4, 20 and 10, &c., will give a mixture of the same value.

QUEST.-251. What is Alligation Alternate? How do you prove Alligation Alternate ?

CASE I.

252. To find the proportion in which several simples of given prices must be mixed together, that the compound may be worth a given price.

I. Set down the prices of the simples one under the other, in the order of their values, beginning with the lowest.

II. Link the least price with the greatest, and the one next to the least with the one next to the greatest and so on, until the price of each simple which is less than the price of the mixture is linked with one or more that is greater; and every one that is greater with one or more that is less.

III. Write the difference between the price of the mixture and that of each of the simples opposite that price with which the particular simple is linked; then the difference standing opposite any one price, or the sum of the differences when there is more than one, will express the quantity to be taken of that price.

EXAMPLES.

1. A merchant would mix wines worth 16s., 18s., and 22s. per gallon in such a way, that the mixture may be worth 20s per gallon: how much must be taken of each sort?

Ans.

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2gal. at 16s., 2 at 18s., and 6 at 22s.: or any other quantities bearing the proportion of 2, 2, and 6.

2. What proportions of coffee at 8cts., 10cts., and 14cts. per lb. must be mixed together so that the compound shall be worth 12cts. per lb. ?

3. A goldsmith has gold of 16, of 18, of 23, and of 24 carats fine: what part must be taken of each so that the mixture shall be 21 carats fine?

QUEST.--252. How do you find the proportions so that the compound may be of a given price?

4. What portion of brandy at 14s. per gallon, of old Madeira at 24s. per gallon, of new Madeira at 21s. per gallon, and of brandy at 10s. per gallon, must be mixed together so that the mixture shall be worth 18s. per gallon ?

CASE II.

253. When a given quantity of one of the simples is to be taken.

I. Find the proportional quantities of the simples as in

Case I.

II. Then say, as the number opposite the simple whose quantity is given, is to either proportional quantity, so is the given quantity, to the proportional part of the corresponding simple.

EXAMPLES.

1. How much wine at 5s., at 5s. 6d., and 6s. per gallon must be mixed with 4 gallons at 4s. per gallon, so that the mixture shall be worth 5s. 4d. per gallon?

Ans. 1gal. at 5s., 2 at 5s. 6., and 8 at 6s

QUEST.-253. How do you find the proportion when the quantity of one of the simples is given?

2. A farmer would mix 14 bushels of wheat at $1,20 per bushel, with rye at 72ċts., barley at 48cts., and oats at 36cts. : how much must be taken of each sort to make the mixture worth 64 cents per bushel?

3. There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., with 12 bushels of oats at 18d. per bushel: how much has been taken of each sort when the mixture is worth 3s. 6d. ?

4. A distiller would mix 40gal. of French brandy at 12s. per gallon, with English at 7s. and spirits at 4s. per gallon : what quantity must be taken of each sort, that the mixture may be afforded at 8s. per gallon?

CASE III.

254. When the quantity of the compound is given as wel as the price.

I. Find the proportional quantities as in Case I.

II. Then say, as the sum of the proportional quantities, is to each proportional quantity, so is the given quantity, to the corresponding part of each.

EXAMPLES.

1. A grocer has four. sorts of sugar worth 12d., 10d., 6d., and 4d. per pound; he would make a mixture of 144lb. worth 8d. per pound: what quantity must be taken of each sort?

QUEST.-254. How do you determine the proportion when the quantity

of the compound is given as well as the price?

Hence, the average cost is 8d.

2. A grocer having four sorts of tea worth 5s., 6s., 8s., and 9s. per lb., wishes a mixture of 871b. worth 7s. per lb. : how much must be taken of each sort?

3. A vintner has four sorts of wine, viz., white wine at 4s. per gallon, Flemish at 6s. per gallon, Malaga at 8s. per gallon, and Canary at 10s. per gallon: he would make a mixture of 60 gallons to be worth 5s. per gallon: what quantity must be taken of each?

4. A silversmith has four sorts of gold, viz., of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine: he would make a mixture of 42oz. of 17 carats fine: how much must be taken of each sort?

CUSTOM HOUSE BUSINESS.

255. PERSONS Who bring goods, or merchandise, into the United States, from foreign countries, are required to land them at particular places or ports, called Ports of Entry, and to pay a certain amount on their value, called a Duty. This duty is imposed by the General Government, and must be the same on the same articles of merchandise, in every part of the United States.

Besides the duties on merchandise, vessels employed in commerce are required, by law, to pay certain sums for the privilege of entering the ports. These sums are large or

QUEST.-255. What is a port of entry? What is a duty? By whom are duties imposed? What charges are vessels required to pay? What are the moneys arising from duties and tonnage called?